Parallelizing neural networks during training

ABSTRACT

A parallel convolutional neural network is provided. The CNN is implemented by a plurality of convolutional neural networks each on a respective processing node. Each CNN has a plurality of layers. A subset of the layers are interconnected between processing nodes such that activations are fed forward across nodes. The remaining subset is not so interconnected.

TECHNICAL FIELD

The following relates generally to convolutional neural networks and more specifically to a parallel convolutional neural network.

BACKGROUND

Convolutional neural networks (CNNs) are powerful models that can be configured to be well suited for computer vision problems. CNNs typically perform best when they are large (i.e., more complex), meaning that they have more, deeper and highly-interconnected layers. A primary drawback of these CNNs is computational cost. Thus, large CNNs are typically impractically slow. In many applications, a large CNN demands more computation than is currently available on a serial computer.

Complex CNNs can, therefore, be implemented by parallelizing the network across multiple processors. For example, for image processing or classification tasks, a CNN could be implemented on several graphics processing units (GPUs).

There have been various proposals to increase the practicality of CNNs by means of parallelizing the CNN across several processors. Such approaches partition the network into parallel subnetworks in such a way as to minimize communication cost.

A first approach naively partitions the network into parallel subnetworks and communicates the state of every subnetwork's layer to all other subnetworks. This approach may be applied to both CNNs and fully connected networks.

In this approach, the network is partitioned into some number of parallel subnetworks in some manner. The subnetworks communicate their activations to all other subnetworks at every layer, which results in a parallel implementation of the feedforward neural network.

In certain implementations, however, this approach is inefficient with CNNs. Its efficiency is best for fully connected weight matrices because the amount of computation required by such matrices causes the communication-to-computation ratio to be small.

In contrast, CNN weight matrices are much sparser, so their communication-to-computation ratio is much larger. As a result, when applying this approach to CNNs, a large fraction of time is spent on communication, which makes the parallelization less useful.

A second approach partitions the network into slices that communicate with their neighbors, and is commonly only applied to convolutional or to locally connected networks. However, current implementations of this approach typically handle pooling inefficiently. Pooling is a technique for making the network's activations more invariant to small translations. While pooling increases the accuracy of the CNN, it changes the dimensions of the activation tensor in such a way that typically allows for less parallelism and requires increased communication for the second approach.

For example, one particular implementation of the second approach parallelizes CNNs into slices that communicate only with their neighbours. The approach partitions the input tensor (of size N×N×u) into m subtensors of size (N/m)×N×u and allocates a computing node to each of the m subtensors. This is efficient only when N is large and u is small, because a large N allows m, and thus the number of computing nodes, to be large, and a small u allows the neighbouring slices to not communicate much. However, when pooling is used, N is necessarily small and u is necessarily large. Since m cannot exceed N, a small N restricts the number of computing nodes which limits the attainable acceleration, while a large u requires more communication between neighbouring slices which increases the communication cost.

It is an object of the following to obviate or mitigate at least one of the foregoing issues.

SUMMARY

In one aspect, a parallel convolutional neural network is provided, the parallel convolutional neural network comprising a plurality of convolutional neural networks each implemented on a respective processing node and each comprising a plurality of layers, a subset of said layers being interconnected between said processing nodes to feed forward respective activations and a remaining subset not being so interconnected.

In another aspect, a method for establishing a parallel convolutional neural network is provided.

DESRIPTION OF THE DRAWINGS

The features of the invention will become more apparent in the following detailed description in which reference is made to the appended drawings wherein:

FIG. 1 is a four-layer feedforward neural network;

FIG. 2 is a parallel convolutional neural network;

FIG. 3 is a flowchart depicting a method of establishing a parallel convolutional neural network; and

FIG. 4 is a neural network comprising a parallel convolutional neural network.

DETAILED DESCRIPTION

Embodiments will now be described with reference to the figures. It will be appreciated that for simplicity and clarity of illustration, where considered appropriate, reference numerals may be repeated among the figures to indicate corresponding or analogous elements. In addition, numerous specific details are set forth in order to provide a thorough understanding of the embodiments described herein. However, it will be understood by those of ordinary skill in the art that the embodiments described herein may be practiced without these specific details. In other instances, well-known methods, procedures and components have not been described in detail so as not to obscure the embodiments described herein. Also, the description is not to be considered as limiting the scope of the embodiments described herein.

It will also be appreciated that any module, unit, component, server, computer, terminal or device exemplified herein that executes instructions may include or otherwise have access to computer readable media such as storage media, computer storage media, or data storage devices (removable and/or non-removable) such as, for example, magnetic disks, optical disks, or tape. Computer storage media may include volatile and non-volatile, removable and non-removable media implemented in any method or technology for storage of information, such as computer readable instructions, data structures, program modules, or other data. Examples of computer storage media include RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by an application, module, or both. Any such computer storage media may be part of the device or accessible or connectable thereto. Any application or module herein described may be implemented using computer readable/executable instructions that may be stored or otherwise held by such computer readable media.

The following describes a system and method for parallelizing a neural network in which a parallel neural network comprises a plurality of neural networks that communicate to other ones of the neural networks a subset of their state conditions. Thus, communications cost is relatively low compared to a fully connected parallel neural network, while maintaining suitable ability to benefit from parallel computation. Pooling is also enabled under the described system and method.

The following describes the parallelization of a CNN, however it is to be understood that the following principles apply to neural networks generally.

Referring first to FIG. 1, a feedforward neural network (FNN) comprises a plurality of layers, each of which comprises a pluralirty of cells. In various implementations, the network, the layers and/or the cells are each implemented by a processor.

An FNN is a family of functions that map

to

which is parameterized with a sequence of weight matrices (W₁, . . . , W_(l)) and a sequence of bias vectors (b₁, . . . , b_(l)). Given an input xε

, the network's output x_(l) can be computed by x_(i)←f_(i)(W_(i)x_(i-1)+b_(i)) by iterating i from 1 to l, where x₀←x, x_(i) is the activation vector (or state) of the i-th layer, and f_(i)(•) is an easy to compute nonlinearity, such as the element-wise tan h or the element-wise rectification max(0,x).

Every configuration of weights and biases (W₁, . . . , W_(l)) and (b₁, . . . , b_(l)) defines a different x₀→x_(l) mapping. Training the network comprises applying nonlinear optimization methods to find a parameter setting whose mapping x₀→x_(l) produces correct labels on the training set. An FNN is said to be fully-connected when each of the W_(i) matrices is dense.

Neural networks can, in principle, be applied to vision tasks if the input image is encoded as a vector. However, even small images are extremely high-dimensional objects. For example, a 256×256 color image has 196608 dimensions. This high dimensionality typically renders neural networks infeasible to apply in practice, because their weight matrices tend to be impractically large. For instance, a dense square matrix of the above dimensions has almost 5·10¹⁰ parameters, which requires more computation than is typically available on a single CPU. It also typically requires an infeasible amount of data for learning.

A CNN, however, typically requires much less computation and training data than a fully-connected neural network with similarly-sized layers, while being capable of achieving similar performance on tasks such as vision (provided there are no limits on the amount of data and computation). The CNN has a restricted connectivity, so each neuron is only connected to a small patch of the input image as opposed to the entire image, which subsequently reduces the number of connections. This restriction does not hurt performance relative to a fully-connected network, because weighted sums of spatially-separated pixels are not informative. In addition, the CNN uses weight sharing, which means that it processes every image patch with the same connections. This results in an additional reduction in the number of parameters, which greatly reduces the amount of necessary training data. This weight sharing does not hurt performance relative to networks that do not use such sharing, because image statistics are homogeneous, so images can be reasonably processed in the same manner in every locations.

Stated more formally, the input I to a CNN weight matrix may be a stack of u images of size N×N each, which is a tensor of size N×N×u. A single CNN layer may apply u×v filters, each of size (2 m+1)×(2 m+1), to produce an output tensor (activation tensor) O of size (N−2 m)×(N−2 m)×v. The output tensor may be computed with the following formula for each i,j, and v′:

$O_{i,j,v^{\prime}} = {\sum\limits_{x = {- m}}^{m}{\sum\limits_{y = {- m}}^{m}{\sum\limits_{u^{\prime} = 0}^{u}{I_{{i^{\prime} + x},{j^{\prime} + y},u^{\prime}}W_{x,y,u^{\prime},v^{\prime\;}}}}}}$ where i′=i+m+1 and j′=j+m+1.

Thus, while a fully connected network with identically-sized layers has N²u×(N−2 m)²v connections, a convolutional neural networks has (N−2 m)²u×(2 m+1)²v connections and only u×v×(2 m+1)² parameters, so the number of connections is reduced by a factor of about N²/4 m² and the number of parameters is reduced by a factor of almost N⁴/4 m².

However, despite the relative efficiency of a reasonably-sized convolutional neural network, in practice it may be impracticably slow for classification of large images.

Referring now to FIG. 2, in one aspect, a system for parallelizing a CNN is provided. The system comprises a plurality of CNNs instantiated on a plurality of computing nodes. Each computing node is a processor such as a CPUs or GPUs. It will be appreciated that a set of nodes may comprise combinations of CPUs and GPUs as well as other processors. It will also be appreciated that the described CNN need not be applied only to image processing, but can be applied to other suitable tasks.

In one aspect, the system comprises interconnections initiated at a predetermined subset of layers for which activations will be communicated to other CNNs. The activations may be communicated to the subsequent adjacent layer of the other CNNs. For example, activations of nodes at layer i are communicated to cells of layer i+1 in other nodes. In the example shown in FIG. 2, for example, activations of layer 2 and 4 in each node are communicated to layer 3 and 5, respectively, of the other nodes.

The layers selected for interconnection are a subset of all layers. In an example, which is to be understood as non-limiting, activations may be communicated across all nodes of particular pair of adjacent layers at predetermined intervals (i.e., nodes of layer xi+k are communicated to nodes of layer xi+k+1, where x is an integer and k is an offset constant to define the first such interconnected layer). In a specific example, the selected layers are every third or fourth layer (i.e., x=3 or 4). In another example, the interval of such layers is irregular, such that the layers whose activations are to be communicated are selected arbitrarily, or selected based on additional considerations.

In another aspect, activations of a particular node may be communicated to a subset of the other nodes. For example, when the number of computing nodes is large, such as being greater than 10 for example, the cost of communicating the activation of every CNN at the predetermined layers to each other CNN at the respective subsequent layers may be impractically or prohibitively expensive. In such a case, the activations may be communicated to a predetermined subset (that may be selected randomly or in some other way prior to training) of the other CNNs. In an example, activations for node 1 layer 1 may be interconnected to node 2 layer 2 but not node 3 layer 2.

In yet another aspect, activations of nodes of a particular layer may be interconnected with the subsequent layers of other CNNs while nodes in the corresponding layer of the other CNNs are not interconnected with the subsequent layers of other CNNs. In an example, node 1 layer 1 may be interconnected to node 2 layer 2 and node 3 layer 2, while node 2 layer 1 may only be connected to node 2 layer 2 and not interconnected to node 1 layer 2 or node 3 layer 2. In another example, layer 1 of both nodes 1 and 2 may be interconnected with layer 2 of each node, while layer 1 of node 3 may only be connected to node 3 layer 2.

Combinations of the foregoing are also possible. For example, node 1 layer 1 may be interconnected to node 2 layer 2 but not node 3 layer 2; while node 2 layer 1 may not be interconnected to either node 1 layer 2 or node 3 layer 2; and node 3 layer 1 may be interconnected to both node 1 layer 2 and node 2 layer 2.

Such examples may be beneficial where, for example, one or more node (one or more of the CPUs) is slower than others. In this example, it may be beneficial to reduce the size of each layer at the slower nodes relative to that layer at the other nodes to enable all the CPUs to process each layer at roughly the same speed. In turn, the smaller layer size of said GPU may make it cheap to communicate at every layer, without greatly increasing communication cost. It may thus be cost-effective to communicate the activations of the slower GPU at every layer, while communicating the activations of the faster CPUs at only some of the layers.

Referring to FIG. 3, a method of establishing a parallel CNN may comprise establishing a plurality of parallel CNNs each instantiated on a computing node (300), selecting one or more layers for each CNN for which to communicate their activations to other CNNs (302), selecting, for each such layer in each such CNN, at least a subset of the other CNNs to which to communicate the activations (304), and interconnecting each of the selected layers with the respective subsequent layers of such subset (308).

Referring to FIG. 4, a neural network comprising another example of a parallel CNN implemented on two nodes is shown. In this example, the neural network comprises eight layers where the first five are convolutional and the remaining three are fully-connected. The second, fourth and fifth convolutional layers are connected to the previous layer only within the same processor, while those of the remaining layers are interconnected across the two nodes.

Although the invention has been described with reference to certain specific embodiments, various modifications thereof will be apparent to those skilled in the art without departing from the spirit and scope of the invention as outlined in the claims appended hereto. The entire disclosures of all references recited above are incorporated herein by reference. 

The invention claimed is:
 1. A system for parallelizing a neural network having a plurality of parameters during training of the neural network on a training set to determine a final parameter setting for the parameters of the neural network that produces correct classifications for the training set, the system comprising: a plurality of parallel neural networks, wherein each of the plurality of parallel neural networks is implemented on a respective computing node, and wherein the plurality of parallel neural networks each receive a same input image from the training set and collectively generate an output that classifies the input image, wherein each of the neural networks comprises a respective plurality of layers, wherein each plurality of layers comprises an interconnected layer and a non-interconnected layer, and wherein processing data through the layers of each of the plurality of parallel neural networks comprises: providing output from the interconnected layer to at least one layer in each of the other parallel neural networks in the plurality of parallel neural networks; and providing output from the non-interconnected layer only to a layer of the same parallel neural network, and wherein the system is configured to: after the training, store the final parameter setting for the parameters of the neural network on one or more non-transitory computer storage media.
 2. The system of claim 1, wherein processing data through the layers of each of the plurality of parallel neural networks further comprises: providing output from the interconnected layer to at least one layer of the same parallel neural network.
 3. The system of claim 1, wherein each of the plurality of layers comprises a respective plurality of nodes and wherein each node generates a respective output activation based on an input activation received from one or more other layers.
 4. The system of claim 3, wherein providing output from the interconnected layer to at least one layer of at least one different parallel neural network of the plurality of parallel neural networks comprises providing output activations from each node of the interconnected layer to each node in at least one layer of a subset of the other parallel neural networks in the plurality of parallel neural networks.
 5. The system of claim 3, wherein providing output from the interconnected layer to at least one layer of at least one different parallel neural network of the plurality of parallel neural networks comprises providing output activations from only a subset of the nodes of the interconnected layer to a subset of the nodes in at least one layer of at least one different parallel neural network in the plurality of parallel neural networks.
 6. The system of claim 1, wherein the parallel neural networks are convolutional neural networks.
 7. The system of claim 1, wherein the non-interconnected layer is a convolutional layer.
 8. The system of claim 1, wherein the interconnected layer is a fully-connected layer.
 9. A method for parallelizing a neural network having a plurality of parameters during training of the neural network on a training set to determine a final parameter setting for the parameters of the neural network that produces correct classifications for the training set, the method comprising: processing data using each of a plurality of parallel neural networks, wherein each of the plurality of parallel neural networks is implemented on a respective computing node, wherein the plurality of parallel neural networks each receive a same input image from the training set and collectively generate an output that classifies the input image, wherein each of the neural networks comprises a respective plurality of layers, wherein each plurality of layers comprises an interconnected layer and a non-interconnected layer, wherein processing data using each of the plurality of parallel neural networks comprises processing the data through the layers of each of the plurality of parallel neural networks, and wherein processing the data through the layers of each of the plurality of parallel neural networks comprises: providing output from the interconnected layer to at least one layer in each of the other parallel neural networks in the plurality of parallel neural networks; providing output from the non-interconnected layer only to a layer of the same parallel neural network; and after the training, storing the final parameter setting for the parameters of the neural network on one or more non-transitory computer storage media.
 10. The method of claim 9, wherein processing data through the layers of each of the plurality of parallel neural networks further comprises: providing output from the interconnected layer to at least one layer of the same parallel neural network.
 11. The method of claim 9, wherein each of the plurality of layers comprises a respective plurality of nodes and wherein each node generates a respective output activation based on an input activation received from one or more other layers.
 12. The method of claim 11, wherein providing output from the interconnected layer to at least one layer of at least one different parallel neural network of the plurality of parallel neural networks comprises providing output activations from each node of the interconnected layer to each node in at least one layer of a subset of the other parallel neural networks in the plurality of parallel neural networks.
 13. The method of claim 11, wherein providing output from the interconnected layer to at least one layer of at least one different parallel neural network of the plurality of parallel neural networks comprises providing output activations from only a subset of the nodes of the interconnected layer to a subset of the nodes in at least one layer of at least one different parallel neural network in the plurality of parallel neural networks.
 14. The method of claim 9, wherein the parallel neural networks are convolutional neural networks.
 15. The method of claim 9, wherein the non-interconnected layer is a convolutional layer.
 16. The method of claim 9, wherein the interconnected layer is a fully-connected layer.
 17. A non-transitory computer storage medium encoded with instructions that, when executed by one or more computers, cause the one or more computers to perform operations for parallelizing a neural network having a plurality of parameters during training of the neural network on a training set to determine a final parameter setting for the parameters of the neural network that produces correct classifications for the training set, the operations comprising: processing data using each of a plurality of parallel neural networks, wherein each of the plurality of parallel neural networks is implemented on a respective computing node, wherein the plurality of parallel neural networks each receive a same input image from the training set and collectively generate an output that classifies the input image, wherein each of the neural networks comprises a respective plurality of layers, wherein each plurality of layers comprises an interconnected layer and a non-interconnected layer, wherein processing data using each of the plurality of parallel neural networks comprises processing the data through the layers of each of the plurality of parallel neural networks, and wherein processing the data through the layers of each of the plurality of parallel neural networks comprises: providing output from the interconnected layer to at least one layer in each of the other parallel neural networks in the plurality of parallel neural networks; providing output from the non-interconnected layer only to a layer of the same parallel neural network; and after the training, storing the final parameter setting for the parameters of the neural network on one or more non-transitory computer storage media.
 18. The non-transitory computer storage medium of claim 17, wherein processing data through the layers of each of the plurality of parallel neural networks further comprises: providing output from the interconnected layer to at least one layer of the same parallel neural network.
 19. The non-transitory computer storage medium of claim 17, wherein each of the plurality of layers comprises a respective plurality of nodes and wherein each node generates a respective output activation based on an input activation received from one or more other layers.
 20. The non-transitory computer storage medium of claim 19, wherein providing output from the interconnected layer to at least one layer of at least one different parallel neural network of the plurality of parallel neural networks comprises providing output activations from each node of the interconnected layer to each node in at least one layer of a subset of the other parallel neural networks in the plurality of parallel neural networks.
 21. The non-transitory computer storage medium of claim 19, wherein providing output from the interconnected layer to at least one layer of at least one different parallel neural network of the plurality of parallel neural networks comprises providing output activations from only a subset of the nodes of the interconnected layer to a subset of the nodes in at least one layer of at least one different parallel neural network in the plurality of parallel neural networks.
 22. The non-transitory computer storage medium of claim 17, wherein the parallel neural networks are convolutional neural networks.
 23. The non-transitory computer storage medium of claim 17, wherein the non-interconnected layer is a convolutional layer.
 24. The non-transitory computer storage medium of claim 17, wherein the interconnected layer is a fully-connected layer. 